CUBO A Mathematical Journal
Vol.11, No¯ 03, (25–40). August 2009

Some General Theorems on Uniform Boundedness for

Functional Differential Equations

Tingxiu Wang

Department of Computer Science, Mathematics and Physics,

Missouri Western State University, 4525 Downs Drive,

Saint Joseph, Missouri 64507

email: twang1@missouriwestern.edu

ABSTRACT

Consider the functional differential equation with bounded delay

X
′
= F (t, Xt), X ∈ Rn.

We discuss uniform boundedness and uniform ultimate boundedness by Liapunov’s

second method with conditions such as:

(i) W1(|X(t)|) ≤ V (t, Xt) ≤ W2(|X(t)| +
∫ t

t−h
D(u, Xu)du);

(ii) V (t, φ) ≤ W3(‖φ‖);
(iii) V ′

(1)
(t, Xt) ≤ −γ1(t)W4(m(Xt)) − γ2(t)W5(

∫ t

t−h
D(u, Xu)du) + M ;

where m(φ) = min−h≤s≤0 |φ(s)|.
The theorem discussed in this paper generalizes some results on uniform boundedness

and uniform ultimate boundedness for functional differential equations with bounded

delay. Some examples are also discussed in this paper.



26 Tingxiu Wang CUBO
11, 3 (2009)

RESUMEN

Considere la ecuación diferencial con retardo acotado

X
′
= F (t, Xt), X ∈ Rn.

Discutimos acotamiento uniforme y acotamiento uniforme definitivo mediante el se-

gundo metodo de Liapunov con las condiciones:

(i) W1(|X(t)|) ≤ V (t, Xt) ≤ W2(|X(t)| +
∫ t

t−h
D(u, Xu)du);

(ii) V (t, φ) ≤ W3(‖φ‖);

(iii) V ′
(1)

(t, Xt) ≤ −γ1(t)W4(m(Xt)) − γ2(t)W5(
∫ t

t−h
D(u, Xu)du) + M ;

donde m(φ) = min−h≤s≤0 |φ(s)|.

El teorema discutido en este artículo generaliza algunos resultados de acotamiento

uniforme y acotamiento uniforme definitivo para ecuaciones diferenciales funcionales

con retardo acotado. Algunos ejemplos son presentados.

Key words and phrases: Uniform boundedness, stability, Liapunov’s second method, functional

differential equations.

Math. Subj. Class.: 34D20, 34D40, 34K20.

1 Introduction

We consider the system

X
′
(t) = F (t, Xt), X ∈ Rn, (1)

where Xt(θ) = X(t + θ) for −h ≤ θ ≤ 0 and h is a positive constant. The following notation and
terminology will be used.

Denote by C the space of continuous functions φ : [−h, 0] → Rn. For φ ∈ C we will use the
norm ‖φ‖ := max |φ(s)|, where | · | is any convenient norm in Rn. Given H > 0, CH denotes the
set of φ ∈ C with ‖φ‖ < H. X′(t) denotes the right-hand derivative at t if it exists and is finite.
It is supposed that F : R+ × C → Rn, that F is continuous, and that F takes bounded sets into
bounded sets. Here, R+ = [0, ∞). Then it is known [2, 6, 7, 15] that for each t0 ∈ R+ and each
φ ∈ C there is at least one solution X(t0, φ) of (1) satisfying Xt0 (t0, φ) = φ defined on an interval
[t0, t0 + α) for some α > 0 and if there is an H1 < H with |X(t, t0, φ)| ≤ H1, then α = ∞.

By means of Liapunov’s second method, throughout this paper we work with wedges, denoted

by Wi : R+ → R+, which are continuous and strictly increasing. We also work with continuous



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Some General Theorems on Uniform Boundedness ... 27

functionals V : R+ × C → R+ (called Liapunov functionals) with V (t, 0) ≡ 0, whose derivative V ′
with respect to (1) is defined by

V
′

(1)
(t, φ) = lim

δ→0+
sup[V (t + δ, Xt+δ(t, φ)) − V (t, φ)]/δ.

Definition 1.1. Solutions of (1) are uniformly bounded (U.B.) if for each B1 > 0 there exists

B2 > 0 such that [t0 ≥ 0, ‖φ‖ ≤ B1, t ≥ t0] imply that |X(t, t0, φ)| < B2. Solutions of (1) are
uniformly ultimately bounded (U.U.B.) for bound B if for each B3 > 0 there exists T > 0 such that

[t0 ≥ 0, ‖φ‖ ≤ B3, t ≥ t0 + T ] imply |X(t, t0, φ)| < B.

Because we are also going to state some stability results, it is necessary to tell the difference

of conditions between stability and boundedness. When we discuss stability, we always assume, in

addition to the above general assumptions:

(i) F : R+ × C → Rn, and F (t, 0) ≡ 0 so that X ≡ 0 is a solution of (1), and is called the
zero solution.

(ii) V : R+ × CH → R+, and V (t, 0) ≡ 0.

(iii) Wi(0) = 0 for each wedge Wi(r).

It is a common idea that stability theory can be generalized in a manner parallel to bounded-

ness theory. But the fact is that the development of boundedness theory is much slower than that

of stability theory. For instance, for the system of ordinary differential equations

X
′
= f (t, X), X ∈ Rn, (2)

where f : R+ × D → Rn continuous, and D ⊂ Rn an open set with 0 ∈ D, two classical results
may be stated as the following:

Theorem 1.1. Let V : R+ × D → R+ be continuous and suppose

(i) W1(|X|) ≤ V (t, X) ≤ W2(|X|),

and

(ii) V ′
(2)

(t, X(t)) ≤ −W3(|X(t)|).

Then X ≡ 0 of (2) is uniformly asymptotically stable.

Theorem 1.2. Let V : R+ × D → R+ be continuous and suppose

(i) W1(|X|) ≤ V (t, X) ≤ W2(|X|), with W1(r) → ∞ as r → ∞,

(ii) V ′
(2)

(t, X(t)) ≤ −W3(|X(t)|) + M, M > 0,.

and

(iii) W3(U ) > M , for some U > 0.

Then the solutions of (2) are U.B. and U.U.B.



28 Tingxiu Wang CUBO
11, 3 (2009)

The parallel results of Theorem 1.1 and Theorem 1.2 for delay equations may be found in

[15;p.190; p.202], and stated as the following.

Theorem 1.3. Let V : R+ × CH → R+ be continuous with

(i) W1(‖φ‖) ≤ V (t, φ) ≤ W2(‖φ‖),

and

(ii) V ′
(2)

≤ −W3(‖φ‖).

Then the zero solution of (1) is uniformly asymptotically stable.

Theorem 1.4. Let V : R+ × C → R+ be continuous with

(i) W1(|φ(0)|) ≤ V (t, φ) ≤ W2(|φ(0)|) + W3(‖φ‖),

(ii) V ′
(1)

(t, Xt) ≤ 0, for |X(t)| large,

(iii) W1(r) − W3(r) → ∞ as r → ∞.

Then solutions of (1) are U.B.

Theorem 1.3 is a direct parallel result of Theorem 1.1 for delay equations. It has not proved

to be useful. For applications, investigators gave the next theorem [15; p.192].

Theorem 1.5. Suppose that F (t, φ) is bounded for φ bounded. Let V : R+×C → R+ be continuous
with

(i) W1(|φ(0)|) ≤ V (t, φ) ≤ W2(|φ(0)|) + W3(‖φ‖2), where ‖ · ‖2 denotes the L2-norm;

(ii) V ′
(1)

(t, Xt) ≤ −W3(|X(t)|).

Then X ≡ 0 is uniformly asymptotically stable.

In 1978, Burton [1] eliminated the condition that F (t, φ) is bounded for φ bounded in Theorem

1.5. Since then, stability theory of this type has been developed very much. In 1989, Burton and

Hatvani [4] gave the following quite general results. Concepts of PIM and IP used below will be

defined in the next section.

Theorem 1.6. Suppose that D, V : R+ × CH → R+ are continuous, η : R+ → R+ is PIM, and
the following conditions are satisfied.

(i) W1(|X(t)|) ≤ V (t, Xt) ≤ W2(|X(t)|) + W3(
∫ t

t−h
D(s, Xs)ds);

(ii) V ′
(1)

(t, Xt) ≤ −η(t)W4(D(t, Xt));

(iii) D(t, φ) ≤ W5(‖φ‖);

(iv) for some K ∈ (0, H) there is a wedge Wk such that [t ∈ R+, u : [−2h, 0] → Rn is
continuous, |u(s)| < K for s ∈ [−2h, 0]] imply

Wk(inf{|u(r)| : −h ≤ r ≤ 0}) ≤
∫

0

−h

D(t + s, us)ds.

Then X = 0 is uniformly asymptotically stable.



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Some General Theorems on Uniform Boundedness ... 29

Theorem 1.7. Let V : R+ × CH → R+ be continuous with

(i) W1(|φ(0)|) ≤ V (t, φ) ≤ W2(‖φ‖);

and

(ii) V ′
(1)

(t, Xt) ≤ −η1W3(|X′(t)|)−η2(t)W4(|X(t)|), where η1 > 0 is a constant, limS→∞
∫ t∗+S

t∗

η2(s)ds = ∞ uniformly with respect to t∗, and there are α > 0, r0 > 0 such that r > r0 implies
W3(r) ≥ αr.

Then X=0 is uniformly asymptotically stable.

In 1991, Wang [10, 11] generalized and unified these two theorems and gave the following

general and yet clean theorem.

Theorem 1.8. Let D, V : R+ × CH → R+ with V continuous and D continuous along the
solutions of (1). Suppose that there are continuous functions η1, η2 : R+ → R+ and that the
following conditions hold:

(i) W1(|X(t)|) ≤ V (t, Xt) ≤ W2(|X(t)| +
∫ t

t−h
D(s, Xs)ds);

(ii) V ′
(1)

(t, Xt) ≤ −γ1(t)W3(m(Xt)) − γ2(t)W4(D(t, Xt)); where γ1ǫ IP(S ) for some S > 0,
γ2ǫ PIM, and m(φ) = min−h≤s≤0 |φ(s)|;

(iii) D(t, φ) ≤ W5(‖φ‖).

Then X = 0 is uniformly asymptotically stable.

The research on stability of this type continues. In 1994, Wang [12] improved Theorem 1.8

with weaker decrescentness. But comparing with stability theory, boundedness theory develops

much more slowly than stability theory does. For U.B. although Theorem 1.4 had been proved

before 1966, a parallel result like Theorem 1.5 without the condition that F is bounded for φ

bounded had not been given until 1986. In 1986, Burton and Zhang [5] showed

Theorem 1.9. Let V : R+ × C → R+ be continuous with

(i) W1(|φ(0)|) ≤ V (t, φ) ≤ W2(|φ(0)|) + W3(
∫

0

−h
W4(|φ(s)|)ds),

(ii) V ′
(1)

(t, Xt) ≤ −W4(|X(t)|) + M, M > 0,

(iii) W1(r), W4(r) → ∞, as r → ∞.

Then solutions of (1) is U.B. and U.U.B.

In 1990, Burton [3] showed

Theorem 1.10. Let V : R+ × C → R+ continuous with

(i) W1(|φ(0)|) ≤ V (t, φ) ≤ W2(|φ(0)|) + W3(‖φ‖2),

(ii) V ′
(1)

(t, Xt) ≤ −W4(‖Xt‖2) + M, M > 0,

(iii) W1(r) → ∞, as r → ∞, W4(U/2) ≥ 12M for some U > 0.

Then solutions of (1) are U.B. and U.U.B.



30 Tingxiu Wang CUBO
11, 3 (2009)

In this paper, we are going to give some general theorems like Theorem 1.8, generalize theorems

like Theorem 1.10, and investigate some examples. One application of uniform boundedness and

ultimate uniform boundedness is to prove the existence of periodic solutions. [2] gives much

discussion on periodic solutions. Makay [8] discussed dissipativeness, which is weaker than U.U.B,

and gave an interesting result on periodic solutions. Based on this paper and [13], the author

examines many common functional differential equations, and obtains not only U.B. and U.U.B.,

but also the existence of periodic solutions. For more examples or applications of this paper and

[13], please see [14].

2 Preliminaries

Definition 2.1. A measurable function γ : R+ → R+ is said to be integrally positive with
parameter α > 0 (IP(α)) if limt→∞ inf

∫ t+α

t
η(s)ds > 0.

That is, η ∈ IP(α) implies that there exist T > 0, and Γ > 0 such that for each t > T ,
∫ t+α

t
η(s)ds ≥ Γ. Thus we also denote IP(α, Γ) = IP(α). This definition is equivalent to the

original one, which can be found in [4]. Now we give a weaker definition than the last one.

Definition 2.2. A measurable function γ : R+ → R+ is said to be partially integrally positive
with parameters α > 0, β > 0, and Γ > 0 (PIP(α, β, Γ)) if there is a sequence {tn}∞1 with
α ≤ tn+1 − tn ≤ β such that

∫ tn+α

tn
η(s)ds ≥ Γ.

Clearly η ∈ IP(α, Γ) implies η ∈ PIP(α, β, Γ) for any β ≥ α.

Lemma 2.1. Let f : R+ → R+ be continuous and G(t) =
∫ t

t−h
f (s)ds. If G(t1) ≥ ε for some

t1 ≥ 2h and ε > 0, then there is a closed interval [a, b] of length h containing t1 in which G(t) ≥ ε/2.

The proof of this lemma, which was originally proved by T. Krisztin, can be found in [9].

3 Main Results

Definition 3.1. A functional D : R+ × C → R+ is said to be continuous along solutions of (1) if
D(t, Xt) is continuous on [t0, ∞) for each solution X(t, t0, φ) of (1) defined on [t0, ∞).

Denote

m(φ) = min
−h≤s≤0

|φ(s)| f or each φ ∈ C.

Theorem 3.1. Let V : R+ × C → R+ be continuous and D : R+ × C → R+ be continuous
along solutions of (1). Let γ1 ∈ PIP(α, β, Γ1) and γ2 ∈ IP(h, Γ2). Denote C = max{β, h}. Let
W1, W2, W3, W4, W5 be wedges with W1(r) → ∞, as r → ∞. Assume that

(i) W1(|X(t)|) ≤ V (t, Xt) ≤ W2(|X(t)| +
∫ t

t−h
D(u, Xu)du);

(ii) V (t, φ) ≤ W3(‖φ‖);



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Some General Theorems on Uniform Boundedness ... 31

(iii) V ′
(1)

(t, Xt) ≤ −γ1(t)W4(m(Xt)) − γ2(t)W5(
∫ t

t−h
D(u, Xu)du) + M ;

(iv) There is a ξ > 0 such that

W4(ξ)Γ1 > 10M C, and W5(ξ/2)Γ2 > 10M C.

Then solutions of (1) are U.B. and U.U.B.

Proof. γ1 ∈ PIP(α, β, Γ1) implies that there is a sequence {tn}∞1 with α ≤ tn+1 − tn ≤ β such
that

∫ tn+α

tn
γ1(u)du ≥ Γ1. γ2 ∈ IP(h, Γ2 ) implies that there is T1 > 0 such that for each t > T1,

∫ t+h

t
γ2(u)du ≥ Γ2.

First, we want to show U.B. That is, for each B1 > 0, there is a B2 > 0 such that [t0 ≥ 0, φ ∈
C, ‖φ‖ < B1 , and t ≥ t0 ] imply |X(t, t0, φ)| < B2. Denote X(t) = X(t, t0, φ).

Let T2 = max{t1, T1}, U = W2(2ξ), and ∆ = max{2W3(B1) + (T2 + 5C)M, U}. Let I0 =
[t0, t0 + T2 + 5C], Ik = [t0 + T2 + 5kC, t0 + T2 + 5(k + 1)C], k = 1, 2, 3, · · · .

Claim I. For each k = 0, 1, 2, · · · , there is a qk ∈ Ik such that V (qk, Xqk ) < ∆.

We use mathematical induction to prove it.

For k = 0, integrating (iii) from t0 to t ∈ I0, we have

V (t, Xt) ≤ V (t0, Xt0 ) + (T2 + 5C)M
≤ W3(B1) + (T2 + 5C)M < ∆. (3)

Clearly, there is a q0 ∈ I0 such that V (q0, Xq0 ) < ∆. In fact, q0 can be any number in I0.
Particularly, we take q0 = t0 + T2 + 5C.

For k = n, assume there is a qn ∈ In such that V (qn, Xqn ) < ∆. We want to show there is a
qn+1 ∈ In+1 such that V (qn+1, Xqn+1 ) < ∆.

If this is false, then V (t, Xt) ≥ ∆ on In+1. It is clear that there is a qnǫIn such that

V (qn, Xqn
) = ∆, and V (t, Xt) ≥ ∆ on [qn, t0 + T2 + 5(n + 1)C]

⋃

In+1.

Then

W2

(

|X(t)| +
∫ t

t−h

D(u, Xu)du

)

≥ V (t, Xt) ≥ ∆ ≥ U (4)

on [qn, t0 + T2 + 5(n + 1)C]
⋃

In+1. Particularly, consider the interval

In+1 = [t0 + T2 + 5(n + 1)C + C, t0 + T2 + 5(n + 1)C + 4C] ⊂ In+1.

(4) implies that either there is some t∗ ∈ In+1 with
∫ t∗

t∗−h

D(u, Xu)du ≥
1

2
W

−1

2
(U ) = ξ



32 Tingxiu Wang CUBO
11, 3 (2009)

or |X(t)| ≥ 1
2
W

−1

2
(U ) = ξ for each t ∈ In+1.

Case I.
∫ t∗

t∗−h
D(u, Xu)du ≥ ξ.

By Lemma 2.1, there are a and b with b − a = h and t∗ ∈ [a, b] such that for each t ∈ [a, b]
∫ t

t−h

D(u, Xu)du ≥
1

2
ξ.

Clearly [a, b] ⊂ In+1.

Integrating (iii) from qn to t0 + T2 + 5(n + 2)C, we have

∆ ≤ V (t0 + T2 + 5(n + 2)C, Xt0+T2+5(n+2)C )

≤ V (qn, Xqn ) −
∫ t0+T2+5(n+2)C

qn

γ2(s)W5

(
∫ s

s−h

D(u, Xu)du

)

+ 10M C

≤ ∆ − W5(
1

2
ξ)

∫ a+h

a

γ2(s)ds + 10M C

≤ ∆ − W5(
1

2
ξ)Γ2 + 10M C

< ∆,

a contradiction.

Case II. |X(t)| ≥ ξ for each t ∈ In+1.

Note that In+1 contains three subintervals of length of C. Therefore In+1 contains at least

three members of {tn}, say s1, s2, and s3 with s1 < s2 < s3. Then integrating (iii) from qn to
t0 + T2 + 5(n + 2)C, we have

∆ ≤ V (t0 + T2 + 5(n + 2)C, Xt0+T2+5(n+2)C )

≤ V (qn, Xqn ) −
∫ t0+T2+5(n+2)C

qn

γ1(u)W4(m(Xu))du + 10M C

≤ ∆ − W4(ξ)
∫ s2+α

s2

γ1(s)ds + 10M C

≤ ∆ − W4(ξ)Γ1 + 10M C
< ∆,

a contradiction.

So there is a qn+1 ∈ In+1 such that V (qn+1, Xqn+1 ) < ∆.

The mathematical induction is complete. Therefore for each k = 0, 1, 2, · · · , there is a qk ∈ Ik
such that V (qk, Xqk ) < ∆.

Now for each t ≥ t0,
V (t, Xt) ≤ ∆ if t ∈ I0 (see(3)),



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Some General Theorems on Uniform Boundedness ... 33

or if t ∈ Ik, k = 1, 2, 3, · · · , (iii) implies

V (t, Xt) ≤ V (qk, Xqk ) + 5M C ≤ ∆ + 5M C if t ≥ qk;

or

V (t, Xt) ≤ V (qk−1, Xqk−1 ) + 10M C ≤ ∆ + 10M C if t < qk.

That is W1(|X(t)|) ≤ V (t, Xt) ≤ ∆ + 10M C for each t ≥ t0. Take B2 = W −11 (∆ + 10M C). This
proves U.B.

Next we are going to prove U.U.B. for bound B. B will be determined at the end of the proof.

That is for each B3 > 0, there exists a T > 0 such that [t0 ≥ 0, ‖φ‖ < B3, t ≥ t0 + T ] imply that
|X(t, t0, φ)| < B. The proof is similar to that of U.B. U and ξ are the same as before.

Let

N = max

{[

W3(B3) + T2M + 5M C

W5(ξ/2)Γ2 − 10M C

]

,

[

W3(B3) + T2M + 5M C

W4(ξ)Γ1 − 10M C

]}

+ 1,

and T3 = 10N C + T2, where [x] denotes the greatest integer function.

Let J0 = [t0 + T2, t0 + T3 + 5C].

Claim II. There is a p0 ∈ J0 such that V (p0, Xp0 ) < U .

We show the claim by contradiction. Assume V (t, Xt) ≥ U for each t ∈ J0. Then for each
t ∈ J0,

W2

(

|X(t)| +
∫ t

t−h

D(u, Xu)du

)

≥ U (5)

Note that J0 can contain 2N + 1 subintervals of length 3C, say,

J0i = [t0 + T2 + 5iC + C, t0 + T2 + 5iC + 4C], i = 0, 1, 2, · · · , 2N.

On each J0i, (5) implies that either there is a u
∗

i ∈ J0i such that
∫ u∗i

u∗
i
−h

D(u, Xu)du ≥ ξ, or
|X(t)| ≥ ξ on J0i.

If the number of these {u∗i } is more than N + 1, by Lemma 2.1, there are ai and bi with
bi − ai = h and u∗i ∈ [ai, bi] such that for each t ∈ [ai, bi]

∫ t

t−h

D(u, Xu)du ≥
1

2
ξ.

Clearly [ai, bi]
⋂

[ai+1, bi+1] = ∅ and [ai, bi] ⊂ J0 for each such i.



34 Tingxiu Wang CUBO
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Integrating (iii) from t0 to t0 + T3 + 5C, we have

0 ≤ V (t0 + T3 + 5C, Xt0+T3+5C))

≤ V (t0, Xt0 ) −
∫ t0+T3+5C

t0

γ2(s)W5

(
∫ s

s−h

D(u, Xu)du

)

ds + M (T3 + 5C)

≤ W3(‖Xt0‖) −
N

∑

i=1

∫ bi

ai

γ2(s)W5

(
∫ s

s−h

D(u, Xu)du

)

ds + M (T3 + 5C)

≤ W3(B3) − W5(ξ/2)Γ2N + 10N M C + T2M + 5M C
≤ W3(B3) − N [W5(ξ/2)Γ2 − 10M C] + T2M + 5M C
< 0, by the choice of N

a contradiction. This means that the number of {u∗i } is less than N + 1. Suppose that there are
more than N + 1 intervals of J0i on which |X(t)| ≥ ξ, say these intervals are J0i, i = 1, 2, 3, · · · ,
N + 1. Clearly, each of these intervals contains at least three members of {tn}, say v1i, v2i, and
v3i with v1i < v2i < v3i. Then integrating (iii) from t0 to t0 + T3 + 5C, we have

0 ≤ V (t0 + T3 + 5C, Xt0+T3+5C )

≤ V (t0, Xt0 ) −
∫ t0+T3+5C

t0

γ1(s)W4(m(Xs))ds + M (T3 + 5C)

≤ W3(‖Xt0‖) −
N

∑

i=1

∫ v2i+α

v2i

γ1(s)W4(m(Xs))ds + M (T3 + 5C)

≤ W3(B3) − W4(ξ)N Γ1 + 10N M C + T2M + 5M C
≤ W3(B3) − N [W4(ξ)Γ1 − 10M C] + T2M + 5M C
< 0, by the choice of N

a contradiction.

Therefore there must be a p0 ∈ J0 such that V (p0, Xp0 ) < U .

Now define Jk = [p0 + 5(k − 1)C, p0 + 5kC] for k = 1, 2, 3, · · · .

Claim III. For each k = 1, 2, 3, · · · , there is a pk ∈ Jk such that V (pk, Xpk ) < U .

We use mathematical induction, again. For k = 1, J1 = [p0, p0 + 5C] and by Claim II, we

obviously can take p1 = p0 with V (p1, Xp1 ) < U .

Assume that for k = n, there is a pn ∈ Jn such that V (pn, Xpn ) < U . We want to show
for k = n + 1, there is a pn+1 ∈ Jn+1 such that V (pn+1, Xpn+1 ) < U . Assume for the sake of
contradiction that V (t, Xt) ≥ U on Jn+1. It is clear that there is a pn ∈ Jn such that

V (pn, Xpn
) = U, and V (t, Xt) ≥ U on [pn, p0 + 5nC]

⋃

Jn+1.

Then

W2

(

|X(t)| +
∫ t

t−h

D(u, Xu)du

)

≥ V (t, Xt) ≥ U (6)



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Some General Theorems on Uniform Boundedness ... 35

on [pn, p0 + 5nC]
⋃

Jn+1. Particularly, consider the interval

Jn+1 = [p0 + 5nC + C, p0 + 5nC + 4C] ⊂ Jn+1.

(6) implies that either there is t∗ ∈ Jn+1 with
∫ t∗

t∗−h
D(u, Xu)du ≥ ξ, or |X(t)| ≥ ξ for each

t ∈ Jn+1.

Case I.
∫ t∗

t∗−h
D(u, Xu)du ≥ ξ.

By Lemma 2.1, there are a and b with b − a = h and t∗ ∈ [a, b] such that for each t ∈ [a, b],
∫ t

t−h
D(u, Xu)du ≥ ξ/2. Clearly [a, b] ∈ Jn+1. Integrating (iii) from pn to p0 + 5(n + 1)C, we have

U ≤ V (p0 + 5(n + 1)C, Xp0+5(n+1)C )

≤ V (pn, Xpn ) −
∫ p0+5(n+1)C

pn

γ2(s)W5

(
∫ s

s−h

D(u, Xu)du

)

ds + 10M C

≤ U − W5(ξ/2)
∫ a+h

a

γ2(s)ds + 10M C

≤ U − W5(ξ/2)Γ2 + 10M C
< U,

a contradiction.

Case II. |X(t)| ≥ ξ for each t ∈ Jn+1.

Note that Jn+1 contains three subintervals of length of C. Therefore Jn+1 contains at least

three members of {tn}, say s1, s2, and s3 with s1 < s2 < s3. Then integrating (iii) from pn to
p0 + 5(n + 1)C, we have

U ≤ V (p0 + 5(n + 1)C, Xp0+5(n+1)C )

≤ V (pn, Xpn ) −
∫ p0+5(n+1)C

pn

γ1(s)W4(m(Xu))du + 10M C

≤ U − W4(ξ)
∫ s2+α

s2

γ1(s)ds + 10M C

≤ U − W4(ξ)Γ1 + 10M C
< U

a contradiction.

So there is a pn+1 ∈ Jn+1 such that V (pn+1, Xpn+1 ) < U .

The mathematical induction is complete. Therefore for each k = 0, 1, 2, · · · , there is a pk ∈ Jk
such that V (pk, Xpk ) < U .

Now for each t ≥ t0 + T3 + 5C, there must be an integer k > 0 such that t ∈ Jk. Then (iii)
implies

W1(|X(t)|) ≤ V (t, Xt) ≤ V (pk, Xpk ) + 5M C ≤ U + 5M C if t ≥ pk;



36 Tingxiu Wang CUBO
11, 3 (2009)

or

W1(|X(t)|) ≤ V (t, Xt) ≤ V (pk−1, Xpk−1 ) + 10M C ≤ U + 10M C if t < pk;

The later case will not happen for k = 1 because of the choice of p1.

Take B = W −1
1

(U + 10M C) and T = T3 + 5C. Then for each t ≥ t0 + T , |X(t)| < B. This
proves U.U.B.

Corollary 3.1. Let V : R+ × C → R+ be continuous with

(i) W1(|X(t)|) ≤ V (t, Xt) ≤ W2(|X(t)| +
∫ t

t−h
|X(s)|pds), where W1(r) → ∞, as r → ∞; and

p > 0 is a constant;

(ii) V ′
(1)

(t, Xt) ≤ −γ(t)W6(
∫ t

t−h
|X(s)|pds)+M , where γ ∈ IP(h, Γ), and M > 0 is a constant;

(iii) there is a ξ > 0 such that min{W6(ξ/2), W6(hξp)}Γ > 20M h.

Then solutions of (1) are U.B. and U.U.B.

Proof. In Theorem 3.1, take D(t, Xt) = |X(t)|p. Condition (ii) implies

V
′

(1)
(t, Xt) ≤ −

1

2
γ(t)W6

(
∫ t

t−h

|X(s)|pds
)

− 1
2
γ(t)W6

(
∫ t

t−h

|X(s)|pds
)

+ M

≤ − 1
2
γ(t)W6 (h(m(Xt))

p
) − 1

2
γ(t)W6

(
∫ t

t−h

|X(s)|pds
)

+ M.

Clearly, γ1(t) =
1

2
γ(t) ∈ PIP(h, h, Γ/2), γ2(t) = 12 γ(t) ∈ IP(h, Γ/2). Take W4(r) = W6(hr

p
), and

W5(r) = W6(r). The other conditions of Theorem 3.1 can be verified easily.

With a little stronger condition, we can state a cleaner corollary.

Corollary 3.2. Let V : R+ × C → R+ be continuous with

(i) W1(|X(t)|) ≤ V (t, Xt) ≤ W2(|X(t)| +
∫ t

t−h
|X(s)|pds), where W1(r) → ∞, as r → ∞; and

p > 0 is a constant;

(ii) V ′
(1)

(t, Xt) ≤ −γ(t)W6(
∫ t

t−h
|X(s)|pds) + M , where γ ∈ IP(h, Γ), M > 0 a constant, and

W6(r) → ∞ as r → ∞.

Then solutions of (1) are U.B. and U.U.B.

Remark. In application, the inequality

V (t, Xt) ≤ W2(|X(t)|) + W7
(

∫ t

t−h

D(u, Xu)du

)

(7)

is more often seen. But Condition (i) of Theorem 3.1 looks cleaner and a little more convenient in

proof. It can be shown that (7) and Condition (i) are equivalent.

Proposition 3.1. (i) If W1 and W2 are two wedges on R+, then there are wedges W∗ and W
∗

such that

W∗(s + t) ≤ W1(s) + W2(t) ≤ W ∗(s + t), f or s, t ∈ R+.



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Some General Theorems on Uniform Boundedness ... 37

(ii) If W is a wedge, then there are wedges W1, W2, W3, and W4 such that

W1(s) + W2(t) ≤ W (s + t) ≤ W3(s) + W4(t), f or each s, t ∈ R+.

Proposition 3.1(i) was proved in the both [9, Proposition 5] and [10, Lemma 2]. But Proposi-

tion 3.1(i) only shows that (7) implies Condition (ii) of Theorem 3.1. To show that Condition (ii)

of Theorem 3.1 implies (7), we need Proposition 3.1(ii).

Proof of Proposition 3.1(ii). Obviously

W (s + t) =
1

2
W (s + t) +

1

2
W (s + t) ≥ 1

2
W (s) +

1

2
W (t).

So take W1(s) =
1

2
W (s), and W2(t) =

1

2
W (t). It is also clear that

W (2s) + W (2t) ≥ W (s + t),

since s + t ≤ max{2s, 2t}. Now take W3(s) = W (2s), and W4(t) = W (2t). This proves Proposition
3.1 (ii).

4 Examples

Example 4A. Consider the scalar equation

x
′
(t) = −a(t)x(t) +

∫ t

t−h

b(s)x(s)ds + f (t, xt) (8)

with a : R+ → R+ and b : [−h, ∞) → R continuous, and f (t, φ) : R+ × C → R continuous.

Theorem 4.1. Suppose that the functions a and b of (8) satisfy:

(a) there is a constant θ > 0 with 0 < θh < 1 such that |b(t)| − θa(t) ≤ 0;

(b)
∫ t

t−h
a(s)ds ≤ B for some constant B > 0, and

∫ t

t−h
a(s)ds is PIP(α, β, Γ) for some

constants α > 0, β > 0, and Γ > 0;

(c) |f (t, φ)| ≤ M for (t, φ) ∈ R+ × C, and M > 0 is a constant.

Then solutions of (8) are U.B. and U.U.B.

Proof. The conclusion follows Theorem 3.1. Find θ0 > 0 and δ > 0 such that θ0 = θ + δ and

0 < θ0h < 1. This can be done, for instance, by taking δ =
1−θh
2h

. Then for t ≥ 0, |b(t)| − θ0a(t) ≤
−δa(t).

Define

V (t, xt) = |x(t)| + θ0
∫

0

−h

∫ t

t+s

a(u)|x(u)|duds



38 Tingxiu Wang CUBO
11, 3 (2009)

and D(t, xt) = a(t)|x(t)|. Then we have

|x(t)| ≤ V (t, xt) ≤ |x(t)| + θ0h
∫ t

t−h

a(u)|x(u)|du

≤ |x(t)| + θ0h
∫ t

t−h

D(s, xs)ds.

Therefore Condition (i) of Theorem 3.1 is satisfied. Clearly,

V (t, φ) ≤
(

1 + θ0h

∫ t

t−h

a(s)ds

)

‖φ‖ ≤ (1 + θ0hB)‖φ‖

by Condition (b). Hence Condition (ii) of Theorem 3.1 is fulfilled.

We also have

V
′
(t, xt) ≤ −a(t)|x(t)| +

∫ t

t−h

|b(u)||x(u)|du + |f (t, xt)|

+ θ0ha(t)|x(t)| − θ0
∫ t

t−h

a(u)|x(u)|du

= (θ0h − 1)a(t)|x(t)| +
∫ t

t−h

[|b(u)| − θ0a(u)]|x(u)|du + M

≤ (θ0h − 1)a(t)|x(t)| − δ
∫ t

t−h

a(u)|x(u)|du + M (9)

≤ − 1
2
δ

∫ t

t−h

a(u)du m(xt) −
1

2
δ

∫ t

t−h

D(u, xu)du + M (10)

This implies that Condition (iii) of Theorem 3.1 is satisfied.

Take W4(r) = r, and W5(r) = r. Then W4(r) → ∞ and W5(r) → ∞ as r → ∞. Therefore
Condition (iv) of Theorem 3.1 is also fulfilled. Now according to Theorem 3.1, solutions of (8) are

U.B. and U.U.B.

Remark: If we use (9), we need to assume a ∈ PIP(α, β, Γ) which is clearly stronger than
∫ t

t−h
a(s)ds ∈ PIP(α, β, Γ). So Condition (iii) of Theorem 3.1 is weaker than

V
′

(1)
(t, Xt) ≤ −γ1(t)W4(|X(t)|) − γ2(t)W5

(
∫ t

t−h

D(u, Xu)du

)

+ M.

Theorem 4.2.Consider Equation (8) again. Suppose that

(a) there are constants k > 1, α > 0, β > 0, and Γ > 0 such that

−a(t) + kh|b(t)| := −γ(t) ≤ 0 and γ ∈ PIP(α, β, Γ);

(b)
∫ t

t−h
|b(s)|ds ≤ B for each t ≥ 0 and some constant B ≥ 0;

(c) |f (t, φ)| ≤ M for some constant M ≥ 0 and each (t, φ) ∈ R+ × C.



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Some General Theorems on Uniform Boundedness ... 39

Then solutions of (8) are U.B. and U.U.B.

Proof. The conclusion follows Theorem 3.1. Define D(t, φ) = |b(t)||φ(0)| and

V (t, xt) = |x(t)| + k
∫

0

−h

∫ t

t+s

|b(u)||x(u)|duds.

Then

V
′
(t, xt) ≤ −a(t)|x(t)| +

∫ t

t−h

|b(s)||x(s)|ds + |f (t, x + t)|

+ k

∫

0

−h

|b(t)||x(t)|ds − k
∫

0

−h

|b(t + s)||x(t + s)|ds

= (−a(t) + kh|b(t)|)|x(t)| + (1 − k)
∫ t

t−h

|b(s)||x(s)|ds + M

= −γ(t)|x(t)| − (k − 1)
∫ t

t−h

D(s, xs)ds + M.

All the other conditions of Theorem 3.1 can be verified easily. Therefore the solutions of (4A) are

U.B. and U.U.B.

Received: January 27, 2008. Revised: March 10, 2008.

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